Generalized Lyapunov–Schmidt Reduction Method and Normal Forms for the Study of Bifurcations of Periodic Points in Families of Reversible Diffeomorphisms

Abstract

We introduce a general reduction method for the study of periodic points near a fixed point in a family of reversible diffeomorphisms. We impose no restrictions on the linearization at the fixed point except invertibility, allowing higher multiplicities. It is shown that the problem reduces to a similar problem for a reduced family of diffeomorphisms, which is itself reversible, but also has an additional ℤ q -symmetry. The reversibility in combination with the ℤ q -symmetry translates to a 𝕋 q -symmetry for the problem, which allows to write down the bifurcation equations. Moreover, the reduced family can be calculated up to any order by a normal form reduction on the original system. The method of proof combines normal forms with the Lyapunov–Schmidt method, and makes repetitive use of the Implicit Function Theorem. As an application we analyze the branching of periodic points near a fixed point in a family of reversible mappings, when for a critical value of the parameters the linearization at the fixed point has either a pair of simple purely imaginary eigenvalues that are roots of unity or a pair of non-semisimple purely imaginary eigenvalues that are roots of unity with algebraic multiplicity 2 and geometric multiplicity 1.